Abstract

A method for automated fringe analysis is presented. It robustly estimates local fringe density and direction in noisy wrapped phase maps. Such information can be used to improve the performance of two-dimensional phase unwrapping methods, to construct phase-jump-preserving filtering strategies, and also to perform robust segmentation of phase data. The method, which is highly insensitive to noise, is model based and performs the estimation in the Fourier domain.

© 2001 Optical Society of America

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References

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  1. D. W. Robinson, G. T. Reid, eds., “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, UK, 1993), pp. 194–228.
  2. K. J. Gåsvik, “Fringe analysis,” in Optical Metrology (Wiley, Chichester, 1995), pp. 267–271.
  3. D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).
  4. P. Stephenson, D. R. Burton, M. I. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
    [CrossRef]
  5. P. Ettl, K. Creath, “Comparison of phase-unwrapping algorithms by using gradient of first failure,” Appl. Opt. 35, 5108–5114 (1996).
    [CrossRef] [PubMed]
  6. O. Marklund, “Noise-insensitive two-dimensional phase unwrapping method,” J. Opt. Soc. Am. A 15, 42–60 (1998).
    [CrossRef]
  7. Z. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893–897 (1996).
    [CrossRef] [PubMed]
  8. A. K. Jain, “The slant transform,” in Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 161–163.
  9. A. W. Oppenheim, R. W. Shaefer, “Fourier analysis of stationary random signals: the periodogram,” in Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 161–163.
  10. A. K. Jain, “Generalized cepstrum and homomorphic filtering,” in Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 259–260.

1998

1996

1994

P. Stephenson, D. R. Burton, M. I. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Burton, D. R.

P. Stephenson, D. R. Burton, M. I. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Creath, K.

Ettl, P.

Gåsvik, K. J.

K. J. Gåsvik, “Fringe analysis,” in Optical Metrology (Wiley, Chichester, 1995), pp. 267–271.

Ghiglia, D. C.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

Jain, A. K.

A. K. Jain, “The slant transform,” in Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 161–163.

A. K. Jain, “Generalized cepstrum and homomorphic filtering,” in Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 259–260.

Lalor, M. I.

P. Stephenson, D. R. Burton, M. I. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Liang, Z.

Z. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893–897 (1996).
[CrossRef] [PubMed]

Marklund, O.

Oppenheim, A. W.

A. W. Oppenheim, R. W. Shaefer, “Fourier analysis of stationary random signals: the periodogram,” in Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 161–163.

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

Shaefer, R. W.

A. W. Oppenheim, R. W. Shaefer, “Fourier analysis of stationary random signals: the periodogram,” in Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 161–163.

Stephenson, P.

P. Stephenson, D. R. Burton, M. I. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Appl. Opt.

IEEE Trans. Med. Imaging

Z. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893–897 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Opt. Eng.

P. Stephenson, D. R. Burton, M. I. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Other

D. W. Robinson, G. T. Reid, eds., “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, UK, 1993), pp. 194–228.

K. J. Gåsvik, “Fringe analysis,” in Optical Metrology (Wiley, Chichester, 1995), pp. 267–271.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

A. K. Jain, “The slant transform,” in Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 161–163.

A. W. Oppenheim, R. W. Shaefer, “Fourier analysis of stationary random signals: the periodogram,” in Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 161–163.

A. K. Jain, “Generalized cepstrum and homomorphic filtering,” in Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 259–260.

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Figures (12)

Fig. 1
Fig. 1

(a) Signal realization tia, (b) signal realization tib.

Fig. 2
Fig. 2

(a) Wrapped phase map ϕα, (b) local neighborhood N1ϕα (upper left corner of ϕα), (c) middle row (N1ϕα)i,32, (d) middle column (N1ϕα)32,j.

Fig. 3
Fig. 3

(a) Neighborhood N2ϕ, (b) neighborhood N3ϕ, (c) neighborhood N4ϕ, (d) neighborhood N5ϕ, (e) estimate  2Pˆ, (f) estimate  3Pˆ, (g) estimate  4Pˆ, (h) estimate  5Pˆ.

Fig. 4
Fig. 4

(a) Row (N1ϕ)i,32, (b) row (N1ϕ˜)i,32, (c) periodogram P{(N1ϕ˜)i,32}, (d) modified periodogram {(N1ϕ˜)i,32}.

Fig. 5
Fig. 5

(a) Modified estimate  2ˆ, (b) modified estimate  3ˆ, (c) modified estimate  4ˆ, (d) modified estimate  5ˆ.

Fig. 6
Fig. 6

(a) Remaining undetermined sections after one subdivision, (b) approximately 1-D neighborhoods.

Fig. 7
Fig. 7

(a) Gray-scale coded map of |τk|{21, 32, 64}, (b) gray-scale coded map of |τk|{16, 21, 32, 64}, (c) Detail of ϕα with direction estimates indicated by white arrows.

Fig. 8
Fig. 8

(a) Result after test run 1(a). (b) Result after test run 1(b).

Fig. 9
Fig. 9

(a) Gray-scale coded map of |τ0|{21, 32, 64}, (b) gray-scale coded map of |τ0|{16, 21, 32, 64}, (c) detail of ϕα with direction estimates indicated by white arrows.

Fig. 10
Fig. 10

(a) Wrapped noisy phase map ϕ˜α, (b) neighborhood N1ϕ˜α, (c) gray-scale coded map of |τ0|{21, 32, 64}, (d) gray-scale coded map of |τ0|{16, 21, 32, 64}.

Fig. 11
Fig. 11

(a) Phase map ϕ˜δ, (b) segmented phase map, (c) gray-scale coded segmented map of p0{±4,±3,±2,±1}, (d) gray-scale coded segmented map of q0{±4,±3,±2,±1}.

Fig. 12
Fig. 12

Gray-scale coded original map p0.

Tables (4)

Tables Icon

Table 1 Error Percentage with Original Estimates and (xm, xm)=(0, 0)

Tables Icon

Table 2 Error Percentage with Modified Estimates and (xm, xm)=(0, 0)

Tables Icon

Table 3 Error Percentage with Original Estimates and (xm, xm)=(1, 1)

Tables Icon

Table 4 Error Percentage with Modified Estimates and (xm, xm)=(1, 1)

Equations (41)

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φi,j=ϕi,j+2πνi,j
ti=2πi2T+1θ(0),ti-k(2T+1)=tiθ(k),-Ti<T,
ti,j=ti,j,
si,j=t[Ri],[Rj],-W(i, j)W0,otherwise,
Ri=i cos(ψ)-j sin(ψ)+i0Rj=i sin(ψ)+j cos(ψ);
si=-2l=1 (-1)ll sin2πl2RT+1i,-WiW,
P{si}p=|Fsi|p2=Pm/l2,p=lM,l=-RT,,RT0,otherwise.
(Fsi,j1)p=exp-ι 2πi0p2N+1(Fsi,j2)p,
Pˆp=12W+1 l=12W+1P{(Nϕ)i,l}p
Qˆq=12W+1 l=12W+1Q{(Nϕ)i,l}q,
χ=p=0Wfχ(Pˆp)
fx(xy)=|y-p0|,xy>TχPm0,otherwise,
χ=q=0Wfχ(Qˆq)
fχ(xy)=|y-q0|,xy>TχQm0,otherwise,
(NϕA)iff((χxm)and(χxm)),
(NϕA);(χ>xm)or(χ>xm).
(NϕA):(χxm)and(χxm),
(p0M)or(q0M).
(NϕA);(χxm)and(χχm),
(p0=M)and(q0=M).
(NϕA);(χxm)and(χxm),
(p0M)or(q0M).
s˜=arctansin(s)+ηQcos(s)+ηI,
ˆp=12W+1 l=12W+1{(Nϕ)i,l}p
Θˆq=12W+1 j=12W+1Θ{(Nϕ)i,l}q
F log(s˜)=F log(s)+F log(1+ζm);
log(x)=log(|x|)+ιπ(1-sgn(x))/Tl,
sgn(x)=1,x>00,x=0-1,x<0
log(|si|)=-2π l=1 γll cos2πl2RT+1i, -WiW,
γl=0RT 1x sinlπRTxdx,
ιπ(1-sgn(si))/Tl=ιπTl-2Tl l=1 1-(-1)ll sin2πl2RT+1i, -WiW,
(F log(si))p=ι πTlδp-l=11-(-1)lTll-γlπlδp+lM-1-(-1)lTll+γlπlδp-lM,
χ=pωfx(ˆp),
Nu0,v0I=(u0ΩXNX  (u0ΩX+1)NX-1,v0ΩYNY  (v0ΩY+1)NY-1),
(0, 0)<(ΩX, ΩY)=(2-k, 2-l)(1, 1),(k, l)(0  log2(NX)/2)×(0  log2(NY)/2)
NΩu0,v0I={(ΩX(u0-12)+12)NX  (ΩX(u0+12)+12)NX-1,(ΩY(ν0-12)+12)NY  (ΩY(ν0+12)+12)NY-1}.
αu,v=|(Pm)u,v-(Qm)u,v|(Pm)u,v+(Qm)u,v,
τm=min(|τk|),
τm=min(|τk|),
ψˆ=arctan(τk/τk)±π/2
βu,v=(Pm)u,v+(Qm)u,v,

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